The problem can be rephrased to avoid the ambiguity. Something like "A hole is bored through a sphere such that the void in the remaining material has the shape of a cylinder 6 inches tall".

I guess the overall idea is anyway to reveal the elegant mathematical result. Wikipedia does a good job of talking clearly about it:

In geometry, the volume of a band of specified height around a sphere—the part that remains after a hole in the shape of a circular cylinder is drilled through the sphere—does not depend on the sphere's radius.

http://en.wikipedia.org/wiki/Napkin_ring_problem